Conditions for $\varinjlim_i\hom(A_i,B)\xrightarrow{}\hom(\varprojlim_i A_i,B)$ to be an isomorphism? I know that there are canonical isomorphisms
$$\varprojlim_i\hom(A_i,B)\xrightarrow{\sim}\hom(\varinjlim_i A_i,B)$$
$$\varprojlim_i\hom(A,B_i)\xrightarrow{\sim}\hom(A,\varprojlim_i B_i),$$
but the maps in the situation where the arrows are reversed need not be isomorphisms. There is a notion of a compact object A, which is defined so that for any nice enough category and nice enough direct system $(B_i)$ the maps
$$\varinjlim_i\hom(A,B_i)\xrightarrow{}\hom(A,\varinjlim_i B_i)$$
are isomorphisms.

In any nice enough category, is there a notion of an object $A$ such that $\varinjlim_i\hom(A_i,B)\xrightarrow{}\hom(\varprojlim_i A_i,B)$ is an isomorphism for any nice enough inverse system $(A_i)$? Or, are there any general conditions I can apply to find if this map is an isomorphism?

 A: This is just a compact object in the opposite category; we might call such a thing a "cocompact object." I'm not aware of any applications or any interesting examples of this. There aren't any nonzero such objects in, say, $\text{Vect}$, for the following reason: any such object $C$ must have the property that $\text{Hom}(-, C)$ sends infinite products to infinite coproducts, since infinite products are a cofiltered limit of finite products and $\text{Hom}(-, C)$ preserves those: this means we must have
$$\text{Hom}(\prod B_i, C) \cong \bigoplus_i \text{Hom}(B_i, C).$$
But in $\text{Vect}$ the LHS will generally be much larger than the RHS (in the sense that there's a natural map from the RHS to the LHS and it's generally not surjective). For example, taking $B_i = C = k$ to be $1$-dimensional, the LHS is the double dual of the RHS, which for infinite-dimensional vector spaces is always strictly larger (in the sense that the double dual map $V \to V^{\ast \ast}$ is never surjective). So $k$ fails to be cocompact. Since, as for compact objects, a retract of a cocompact object is compact, and every nonzero vector space has $k$ as a retract, it follows that no nonzero vector space is cocompact. (These arguments do depend on the axiom of choice though.)
The only positive result I know along these lines is that if the index set is countable and $B_i = C = \mathbb{Z}$ in $\text{Ab}$ then the natural map above is actually an isomorphism; this is due to Specker. But I am pretty sure it's still not true that $\mathbb{Z} \in \text{Ab}$ is cocompact.
